The starting point is that approximations need to be analyzed using asymptotic methods rather than by more standard polynomial expansions. As often happens in computational mathematics, once a phenomenon is understood from a mathematical standpoint, effective algorithms follow. As reviewed in this monograph, we now have at our disposal a number of very effective quadrature methods for highly oscillatory integrals-Filon-type and Levin-type methods, methods based on steepest descent, and complex-valued Gaussian quadrature. Their understanding calls for a fairly varied mathematical toolbox-from classical numerical analysis, approximation theory, and theory of orthogonal polynomials all the way to asymptotic analysis-yet this understanding is the cornerstone of efficient algorithms.
Audience: The text is intended for advanced undergraduate and graduate students, as well as applied mathematicians, scientists, and engineers who encounter highly oscillatory integrals as a critical difficulty in their computations.
Contents : Chapter 1: Introduction; Chapter 2: Asymptotic theory of highly oscillatory integrals; Chapter 3: Filon and Levin methods; Chapter 4: Extended Filon method; Chapter 5: Numerical steepest descent; Chapter 6: Complex-valued Gaussian quadrature; Chapter 7: A highly oscillatory olympics; Chapter 8: Variations on the highly oscillatory theme; Appendix A: Orthogonal polynomials.

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